When we move our vehicle, we "feel" the speed of it. However, when stationary, we still are moving,but moving by quiet a speed. We know that earth moves with X speed, our own solar system moves at Y speed. etc.
Why do we not feel this speed? Even sitting at my home, my mind must be gyrating at an incredible speed.

4 Answers
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In a car, you have a perception of speed because of (a) the "wind" passing by as you rush through the air which is not moving at the same speed as the vehicle, and (b) you perceive the stationary objects nearby as "moving" off into the distance behind.

As the earth moves in its orbit, you don't notice any "wind" from the planet rushing through space, as the atmosphere of the earth is moving along with the planet. There isn't any substance to the space around the planet to create an impression of "wind". The car is surrounded by air, so when you move through the air, you have an impression of moving because of the air rushing past. Because the planet isn't moving through an especially gaseous medium, there is no impression of movement, no "wind".

Also when it comes to our planet, there is nothing "nearby" to create a reference point that would make it appear as if we are moving relative to that reference. Even the closest of objects, such as the moon, are sufficiently far off that the only obvious impression is that these move and we are still. Hence the reason why the geocentric view of our world was dominant for so long.

Even sitting still at home, your entire body is moving at the same velocity as the rest of the planet, so your brain isn't being jostled or 'gyrated' at all. If someone shook you, then your velocity would be changing and your body would feel the effects of acceleration, and then you would be jostled and gyrated. But that is because you are perceiving sudden and rapid changes in velocity, accelerations.

From a (more) physics standpoint our acceleration on earth is basically zero from what we can feel. Just like the others that posted about cars traveling at a certain constant velocity, you won't feel a change. If a car is traveling at a constant velocity there is essentially no feelable force acting on your body. Therefore you you don't feel any effects of the speed in which you are traveling. But on the other hand if you slam on the breaks or accelerate rapidly, you would feel the respected motion on your body (so will the loose items in your car).

In a plane, it's the same thing: you feel the lift off and the altitude rise due to an increase in acceleration to meet a certain altitude in which it's safe to fly. Same with the descent of an airplane: you decelerate to a point in which it's safe to let friction and the planes mechanisms to land timely and in a matter to reach your gate for off boarding.

Another example would be elevators vs. Free falls at a theme park. Most of the time elevators decent without much acceleration but at a somewhat constat velocity so that you don't feel like you're free falling due to gravity (this isn't ALWAYS the feeling we get in elevators though). But if you were to go to a theme park which suspends you in the air and drops you close to the rate of gravity, you know you are plummeting and have that feeling in your stomach.

The earth is the same way as the examples previously described. Same as the others but hopefully in a more explanatory physics way.

You do not change your velocity, unless you fall. The rotation of the earth is constant (gravity ensures you don't leave the surface). The orbit of the earth is constant (essentially in free fall). The sun's orbit in the galaxy is constant (essentially in free fall). The galaxy's path is constant (gravitational influences are too small to measure).
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LDC3Apr 10 '14 at 13:35

If you're big enough and fluid enough, like an atmosphere, you would feel acceleration do to the Earth's motion. Hadley cells are patterns of warm air rising in the tropics, flowing north, and sinking at higher latitudes. The Earth's rotation causes this north-south flow to curve, causing storms to have that distinctive spin you see in in hurricanes.

Alternatively, if you are an exceptionally long pendulum, you'd try to oscillate in the same plane. However, the Earth's rotation would make that plane appearto rotate with respect to the surface of the Earth. Called a Foucault pendulum, it was the first proof the Earth rotates visible from the surface.